EnunciadoÁlgebra Linear e Geometria Analítica - Álgebra Linear - 3ª Edição - J. L. Boldrini- Ed: 3º - Capítulo 3.10 - Ex. 9Encontre A^(- 1) , onde [[[4,-,1,2,-,2],[3,-,1,0,0],[2,3,1,0],[0,7,1,1]]] [[[0,-,i,-,2,i],[1,-,1,i,1],[0,-,1,1,-,i],[1,1,1,0]]] [[[1,0,x],[1,1,x^2],[2,2,x^2]]] MOSTRAR SOLUÇÃO COMPLETAPasso 1 Vamos proceder fazendo operações elementares: [[[4,-,1,2,-,2,|,1,0,0,0],[3,-,1,0,0,|,0,1,0,0],[2,3,1,0,|,0,0,1,0],[0,7,1,1,|,0,0,0,1]]] (l i n h a 1)/(4) : [[[1,-,(1)/(4),(1)/(2),-,(1)/(2),|,(1)/(4),0,0,0],[3,-,1,0,0,|,0,1,0,0],[2,3,1,0,|,0,0,1,0],[0,7,1,1,|,0,0,0,1]]] l i n h a 2 - 3 l i n h a 1 : [[[1,-,(1)/(4),(1)/(2),-,(1)/(2),|,(1)/(4),0,0,0],[0,-,(1)/(4),-,(3)/(2),(3)/(2),|,-,(3)/(4),1,0,0],[2,3,1,0,|,0,0,1,0],[0,7,1,1,|,0,0,0,1]]] l i n h a 3 - 2 l i n h a 1 : [[[1,-,(1)/(4),(1)/(2),-,(1)/(2),|,(1)/(4),0,0,0],[0,-,(1)/(4),-,(3)/(2),(3)/(2),|,-,(3)/(4),1,0,0],[0,(7)/(2),0,1,|,-,(1)/(2),0,1,0],[0,7,1,1,|,0,0,0,1]]] l i n h a 1 - l i n h a 2 : [[[1,0,2,-,2,|,1,-,1,0,0],[0,-,(1)/(4),-,(3)/(2),(3)/(2),|,-,(3)/(4),1,0,0],[0,(7)/(2),0,1,|,-,(1)/(2),0,1,0],[0,7,1,1,|,0,0,0,1]]] l i n h a 3 + 14 l i n h a 2 : [[[1,0,2,-,2,|,1,-,1,0,0],[0,-,(1)/(4),-,(3)/(2),(3)/(2),|,-,(3)/(4),1,0,0],[0,0,-,21,22,|,-,11,14,1,0],[0,7,1,1,|,0,0,0,1]]] l i n h a 4 + 28 l i n h a 2 : [[[1,0,2,-,2,|,1,-,1,0,0],[0,-,(1)/(4),-,(3)/(2),(3)/(2),|,-,(3)/(4),1,0,0],[0,0,-,21,22,|,-,11,14,1,0],[0,0,-,41,43,|,-,21,28,0,1]]] - 4 l i n h a 2 : [[[1,0,2,-,2,|,1,-,1,0,0],[0,1,6,-,6,|,3,-,4,0,0],[0,0,-,21,22,|,-,11,14,1,0],[0,0,-,41,43,|,-,21,28,0,1]]] (l i n h a 3)/(- 21) : [[[1,0,2,-,2,|,1,-,1,0,0],[0,1,6,-,6,|,3,-,4,0,0],[0,0,1,(22)/(- 21),|,(11)/(21),-,(2)/(3),-,(1)/(21),0],[0,0,-,41,43,|,-,21,28,0,1]]] l i n h a 1 - 2 l i n h a 3 : [[[1,0,0,(2)/(21),|,-,(1)/(21),(1)/(3),(2)/(21),0],[0,1,6,-,6,|,3,-,4,0,0],[0,0,1,(22)/(- 21),|,(11)/(21),-,(2)/(3),-,(1)/(21),0],[0,0,-,41,43,|,-,21,28,0,1]]] l i n h a 2 - 6 l i n h a 3 : [[[1,0,0,(2)/(21),|,-,(1)/(21),(1)/(3),(2)/(21),0],[0,1,0,(2)/(7),|,-,(1)/(7),0,(2)/(7),0],[0,0,1,(22)/(- 21),|,(11)/(21),-,(2)/(3),-,(1)/(21),0],[0,0,-,41,43,|,-,21,28,0,1]]] l i n h a 4 + 41 l i n h a 3 : [[[1,0,0,(2)/(21),|,-,(1)/(21),(1)/(3),(2)/(21),0],[0,1,0,(2)/(7),|,-,(1)/(7),0,(2)/(7),0],[0,0,1,(22)/(- 21),|,(11)/(21),-,(2)/(3),-,(1)/(21),0],[0,0,0,(1)/(21),|,(10)/(21),(2)/(3),-,(41)/(21),1]]] l i n h a 1 - 2 l i n h a 4 : [[[1,0,0,0,|,-,1,-,1,4,-,2],[0,1,0,(2)/(7),|,-,(1)/(7),0,(2)/(7),0],[0,0,1,(22)/(- 21),|,(11)/(21),-,(2)/(3),-,(1)/(21),0],[0,0,0,(1)/(21),|,(10)/(21),(2)/(3),-,(41)/(21),1]]] l i n h a 2 - 6 l i n h a 4 : [[[1,0,0,0,|,-,1,-,1,4,-,2],[0,1,0,0,|,-,3,-,4,12,-,6],[0,0,1,(22)/(- 21),|,(11)/(21),-,(2)/(3),-,(1)/(21),0],[0,0,0,(1)/(21),|,(10)/(21),(2)/(3),-,(41)/(21),1]]] l i n h a 3 + 22 l i n h a 4 : [[[1,0,0,0,|,-,1,-,1,4,-,2],[0,1,0,0,|,-,3,-,4,12,-,6],[0,0,1,0,|,11,14,-,43,22],[0,0,0,(1)/(21),|,(10)/(21),(2)/(3),-,(41)/(21),1]]] 21 l i n h a 4 : [[[1,0,0,0,|,-,1,-,1,4,-,2],[0,1,0,0,|,-,3,-,4,12,-,6],[0,0,1,0,|,11,14,-,43,22],[0,0,0,1,|,10,14,-,41,21]]] Portanto, a matriz inversa é: [[[-,1,-,1,4,-,2],[-,3,-,4,12,-,6],[11,14,-,43,22],[10,14,-,41,21]]] Passo 2 Vamos proceder fazendo as operações elementares, veja: [[[0,-,i,-,2,i,|,1,0,0,0],[1,-,1,i,1,|,0,1,0,0],[0,-,1,1,-,i,|,0,0,1,0],[1,1,1,0,|,0,0,0,1]]] t r o q u e d e l u g a r a l i n h a 1 e a l i n h a 2 : [[[1,-,1,i,1,|,0,1,0,0],[0,-,i,-,2,i,|,1,0,0,0],[0,-,1,1,-,i,|,0,0,1,0],[1,1,1,0,|,0,0,0,1]]] l i n h a 4 - l i n h a 1 : [[[1,-,1,i,1,|,0,1,0,0],[0,-,i,-,2,i,|,1,0,0,0],[0,-,1,1,-,i,|,0,0,1,0],[0,2,1,-,i,-,1,|,0,-,1,0,1]]] (l i n h a 2)/(- i) : [[[1,-,1,i,1,|,0,1,0,0],[0,1,-,2,i,,-,1,|,i,0,0,0],[0,-,1,1,-,i,|,0,0,1,0],[0,2,1,-,i,-,1,|,0,-,1,0,1]]] l i n h a 1 + l i n h a 2 : [[[1,0,-,i,0,|,i,1,0,0],[0,1,-,2,i,,-,1,|,i,0,0,0],[0,-,1,1,-,i,|,0,0,1,0],[0,2,1,-,i,-,1,|,0,-,1,0,1]]] l i n h a 3 + l i n h a 2 : [[[1,0,-,i,0,|,i,1,0,0],[0,1,-,2,i,,-,1,|,i,0,0,0],[0,0,1,-,2,i,-,i,-,1,|,i,0,1,0],[0,2,1,-,i,-,1,|,0,-,1,0,1]]] l i n h a 4 - 2 l i n h a 2 : [[[1,0,-,i,0,|,i,1,0,0],[0,1,-,2,i,,-,1,|,i,0,0,0],[0,0,1,-,2,i,-,i,-,1,|,i,0,1,0],[0,0,1,+,3,i,1,|,-,2,i,-,1,0,1]]] (l i n h a 3)/(1 - 2 i) : [[[1,0,-,i,0,|,i,1,0,0],[0,1,-,2,i,,-,1,|,i,0,0,0],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,1,+,3,i,1,|,-,2,i,-,1,0,1]]] l i n h a 1 + i l i n h a 3 : [[[1,0,0,(3 + i)/(5),|,(- 1 + 3 i)/(5),1,(- 2 + i)/(5),0],[0,1,-,2,i,,-,1,|,i,0,0,0],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,1,+,3,i,1,|,-,2,i,-,1,0,1]]] l i n h a 2 + (2 i) l i n h a 3 : [[[1,0,0,(3 + i)/(5),|,(- 1 + 3 i)/(5),1,(- 2 + i)/(5),0],[0,1,0,,(1 + 2 i)/(5),|,(- 2 + i)/(5),0,(- 4 + 2 i)/(5),0],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,1,+,3,i,1,|,-,2,i,-,1,0,1]]] l i n h a 4 + (- 1 - 3 i) l i n h a 3 : [[[1,0,0,(3 + i)/(5),|,(- 1 + 3 i)/(5),1,(- 2 + i)/(5),0],[0,1,0,,(1 + 2 i)/(5),|,(- 2 + i)/(5),0,(- 4 + 2 i)/(5),0],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,0,-,1,|,1,-,i,-,1,1,-,i,1]]] (- 1) l i n h a 4 : [[[1,0,0,(3 + i)/(5),|,(- 1 + 3 i)/(5),1,(- 2 + i)/(5),0],[0,1,0,,(1 + 2 i)/(5),|,(- 2 + i)/(5),0,(- 4 + 2 i)/(5),0],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,0,1,|,-,1,+,i,1,-,1,+,i,-,1]]] l i n h a 1 + (- (3)/(5) - (i)/(5)) l i n h a 4 : [[[1,0,0,0,|,(3)/(5),+,(i)/(5),(2)/(5),-,(i)/(5),(2)/(5),-,(i)/(5),(3)/(5),+,(i)/(5)],[0,1,0,,(1 + 2 i)/(5),|,(- 2 + i)/(5),0,(- 4 + 2 i)/(5),0],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,0,1,|,-,1,+,i,1,-,1,+,i,-,1]]] l i n h a 2 + (- (1)/(5) - (2 i)/(5)) l i n h a 4 : [[[1,0,0,0,|,(3 + i)/(5),(2)/(5),-,(i)/(5),(2)/(5),-,(i)/(5),(3)/(5),+,(i)/(5)],[0,1,0,,0,|,(1)/(5),+,(2 i)/(5),-,(1)/(5),-,(2 i)/(5),-,(1)/(5),+,(3 i)/(5),(1)/(5),+,(2 i)/(5)],[0,0,1,(1)/(5),-,(3 i)/(5),|,-,(2)/(5),+,(i)/(5),0,(1 + 2 i)/(5),0],[0,0,0,1,|,-,1,+,i,1,-,1,+,i,-,1]]] l i n h a 3 + (- (1)/(5) + (3 i)/(5)) l i n h a 4 : [[[1,0,0,0,|,(3)/(5),+,(i)/(5),(2)/(5),-,(i)/(5),(2)/(5),-,(i)/(5),(3)/(5),+,(i)/(5)],[0,1,0,,0,|,(1)/(5),+,(2 i)/(5),-,(1)/(5),-,(2 i)/(5),-,(1)/(5),+,(3 i)/(5),(1)/(5),+,(2 i)/(5)],[0,0,1,0,|,-,(4)/(5),-,(3 i)/(5),-,(1)/(5),+,(3 i)/(5),-,(1)/(5),-,(2 i)/(5),(1)/(5),-,(3 i)/(5)],[0,0,0,1,|,-,1,+,i,1,-,1,+,i,-,1]]] Portanto, a invera da matriz é dada por: [[[(3)/(5),+,(i)/(5),(2)/(5),-,(i)/(5),(2)/(5),-,(i)/(5),(3)/(5),+,(i)/(5)],[(1)/(5),+,(2 i)/(5),-,(1)/(5),-,(2 i)/(5),-,(1)/(5),+,(3 i)/(5),(1)/(5),+,(2 i)/(5)],[-,(4)/(5),-,(3 i)/(5),-,(1)/(5),+,(3 i)/(5),-,(1)/(5),-,(2 i)/(5),(1)/(5),-,(3 i)/(5)],[-,1,+,i,1,-,1,+,i,-,1]]] Passo 3 Vamos proceder como nos itens anteriores: [[[1,0,x,|,1,0,0],[1,1,x^2,|,0,1,0],[2,2,x^2,|,0,0,1]]] l i n h a 2 - l i n h a 1 : [[[1,0,x,|,1,0,0],[0,1,x,(,x,-,1,),|,-,1,1,0],[2,2,x^2,|,0,0,1]]] l i n h a 3 - 2 l i n h a 1 : [[[1,0,x,|,1,0,0],[0,1,x,(,x,-,1,),|,-,1,1,0],[0,2,x^2,-,2,x,|,-,2,0,1]]] l i n h a 3 - 2 l i n h a 2 : [[[1,0,x,|,1,0,0],[0,1,x,(,x,-,1,),|,-,1,1,0],[0,0,- x^2,|,0,-,2,1]]] (l i n h a 3)/(- x^2) : [[[1,0,x,|,1,0,0],[0,1,x,(,x,-,1,),|,-,1,1,0],[0,0,1,|,0,(2)/(x^2),-,(1)/(x^2)]]] l i n h a 1 - x l i n h a 3 : [[[1,0,0,|,1,-,(2)/(x),(1)/(x)],[0,1,x,(,x,-,1,),|,-,1,1,0],[0,0,1,|,0,(2)/(x^2),-,(1)/(x^2)]]] l i n h a 2 - (x^2 - x) l i n h a 3 : [[[1,0,0,|,1,-,(2)/(x),(1)/(x)],[0,1,0,|,-,1,(2 - x)/(x),(x - 1)/(x)],[0,0,1,|,0,(2)/(x^2),-,(1)/(x^2)]]] Disso, a gente tem que a inversa é dada por: [[[1,-,(2)/(x),(1)/(x)],[-,1,(2 - x)/(x),(x - 1)/(x)],[0,(2)/(x^2),-,(1)/(x^2)]]] Resposta [[[-,1,-,1,4,-,2],[-,3,-,4,12,-,6],[11,14,-,43,22],[10,14,-,41,21]]] ; [[[(3)/(5),+,(i)/(5),(2)/(5),-,(i)/(5),(2)/(5),-,(i)/(5),(3)/(5),+,(i)/(5)],[(1)/(5),+,(2 i)/(5),-,(1)/(5),-,(2 i)/(5),-,(1)/(5),+,(3 i)/(5),(1)/(5),+,(2 i)/(5)],[-,(4)/(5),-,(3 i)/(5),-,(1)/(5),+,(3 i)/(5),-,(1)/(5),-,(2 i)/(5),(1)/(5),-,(3 i)/(5)],[-,1,+,i,1,-,1,+,i,-,1]]] ; [[[1,-,(2)/(x),(1)/(x)],[-,1,(2 - x)/(x),(x - 1)/(x)],[0,(2)/(x^2),-,(1)/(x^2)]]] . Ver Outros Exercícios desse livroExercícios de Livros RelacionadosDado um sistema 3X3 com determinante ≠ 0 , calcule o número de multiplicações e divisões necessários para resolvê-lo .a-Pela Regra de Cramer, calculando os determinantes a partir da definição;Dado um sistema 3X3 com determinante ≠ 0 , calcule o número de multiplicações e divisões necessários para resolvê-lo .c-Pelo método da eliminação gaussianaSeja u,v,w vetores linearmente independentes. Mostre que, para quaisquer vetores x,y,z, existe um numero A tal que det(x,y,z)= A.det(u,v,w). Conclua que o det(u,v,w ) ≠ 0 , obtendo assim nova demonstrSeja m e n matrizes tais que o número de colunas de m é igual ao número de linhas de n. prove que ( m . n ) T = n T . m TSeja f uma função que associa a cada matiz 3X3, cujas linhas são u, v e w um numero f(u,v,w) de tal modo que valham as propriedades 1,3 e 4 vistas acima para o determinante. Se f( e 1 ,e 2 ,e 3 )= k, Ver TambémVer Livro Álgebra Linear - 3ª Edição - J. L. BoldriniVer tudo sobre DeterminantesLista de exercícios de Propriedades dos DeterminantesVer exercício 3.10 - 5Ver exercício 9.4 - 8
